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Effect of geometrical symmetry on the angular dependence of the critical magnetic field

in superconductor/normal metal multilayers

S. L. Prischepa,1,2C. Cirillo,2V. N. Kushnir,1E. A. Ilyina,1M. Salvato,2and C. Attanasio2 1State University of Informatics and RadioElectronics, P. Brovka str. 6 Minsk 220013, Belarus 2Dipartimento di Fisica “E.R. Caianiello” and Laboratorio Regionale SuperMat, INFM Salerno,

Università degli Studi di Salerno, Baronissi (Sa), I-84081, Italy

共Received 1 April 2005; published 22 July 2005兲

The angular dependence of the upper critical field, Hc共⌰兲, for Nb/Pd multilayers with different geometrical symmetries共i.e., with an even or an odd number of bilayers, Nbil兲 and thickness of Nb and Pd layers of the order of superconducting coherence length were investigated 共the angle ⌰ is here defined from the plane parallel to the sample surface兲. The presence of an angular dimensional crossover at some temperature depen-dent angle⌰*共T兲 was established. At angles ⌰⬍⌰*共T兲 the multilayers are in two-dimensional mode while at ⌰⬎⌰*共T兲 they are in a three-dimensional one. In addition, the sample with odd N

bilhas a larger value of ⌰*共T兲 with respect to the sample with even N

bil. The experimental results have been qualitatively explained in the framework of the Ginzburg-Landau theory. It was shown that the main features of the Hc共⌰兲 dependences in samples with even and odd Nbilvalues are related to the different localizations of the superconducting order parameter.

DOI:10.1103/PhysRevB.72.024535 PACS number共s兲: 74.78.Fk, 74.20.De, 74.45.⫹c

INTRODUCTION

Considerable effort has been applied to study systems with strong anisotropy. The main reason for these investiga-tions is the unique physical properties of anisotropic objects. Among them layered superconductors have attracted great attention due to the possibility to study the effect of aniso-tropy and the coupling nature between the superconducting layers on the magnetic phase diagram, vortex and pinning properties, superconductivity nucleation, etc.共see, e.g., Ref. 1 for reviews兲. From this point of view the periodic superconductor/normal metal共S/N兲 multilayers prepared by layer-by-layer vacuum deposition are still very interesting systems. The critical parameters of S/N periodic structures differ from the corresponding quantities for thin films and anisotropic superconductors. Varying the layer thickness and the S and N materials it is possible to study both the prox-imity effect between the different metals in coupled multi-layers and the behavior of the lattice of uncoupled supercon-ducting films of different dimensionality.

One of the fundamental quantities characterizing the su-perconducting state is the upper critical magnetic field Hc,

and many efforts have been applied to study its temperature and angular dependences in different S/N systems.2–12In

par-ticular, rather comprehensive understanding of the nature of the dimensional crossover—namely, the transition from three-dimensional共3D兲 to two-dimensional 共2D兲 mode upon a variation of the temperature T—leads to deeper insight into the interaction between the superconducting layers and into the superconducting phase nucleation.13–15Recently,16,17we

have shown that the dimensionality of finite S/N multilayers can be influenced by the “odd-even” effect related to the sample finiteness共in particular, the number of bilayers, odd or even兲 which determines the nucleation position of the su-perconducting phase.18The angular dependence of H

ccan be

of highest importance to get additional useful information

about the structure of multilayers, about the interaction be-tween S layers as well as bebe-tween the adjacent S and N layers, and about the origin of the superconducting phase nucleation in anisotropic superconductors.19

Originally interest in the Hc共⌰兲 study 共⌰ being the angle

between the applied magnetic field and the plane of the sample兲 was pushed by the possibility to be an alternative test of the dimensionality of superconductivity and an addi-tional opportunity to study the influence of the interface boundaries on the properties of the superconducting phase. The fundamental results concerning the Hc共⌰兲 dependences

in homogeneous superconductors were obtained on the basis of the Ginzburg-Landau共GL兲 theory.20–23For infinite

aniso-tropic superconductors the dependence Hc共⌰兲 is a smooth

curve in the whole⌰ range 共including ⌰=0兲. This curve is easily reproduced within the anisotropic effective mass GL theory:23

Hc3D共⌰兲 = Hc共0兲/共cos2⌰ +␥2sin2⌰兲1/2, 共1兲 where ␥= Hc共0兲/Hc共␲/ 2兲 is the anisotropy parameter. For superconductors with two flat surfaces Tinkham derived the formula for the Hc共⌰兲 dependence in the thin-film 2D limit

共dⰆ␰兲,21which gives a discontinuous derivativeH

c共⌰兲/⳵⌰ at⌰=0,

Hc 2D共⌰兲sin ⌰ Hc共␲/2兲

+

Hc 2D共⌰兲cos ⌰ Hc共0兲

2 = 1. 共2兲

It was established also that the Hc共⌰兲 shape and

nucle-ation position of the superconducting phase are mutually re-lated to each other. In the thin-film limit the nucleation al-ways starts at the center of the slab.22 At large sample

thickness the nucleation starts apart from the center and reaches the familiar surface superconductivity situation.20,24

The layered structure of superconducting and nonsuper-conducting films of finite thickness creates additional

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culties for understanding of the angular dependences of the upper critical magnetic field. For example, in S/N multilay-ers in 2D mode关according to the temperature dependence of the parallel critical magnetic field, Hc共T,⌰=0兲兴 the Hc共⌰兲

points fall faster with angle growth with respect to Tinkham’s expression共2兲 for thin films.2,3 Moreover, in the

temperature region of linear Hc共T,⌰=0兲 dependence 共where the 3D mode is realized兲, the angular dependences reveal the pronounced cusp,2,25,26which is a feature of the 2D regime. To resolve these discrepancies between theoretical descrip-tions of homogeneous superconductors and experiments on multilayers, Glazman proposed to take into account both the discrete structure of multilayers and the finite coupling inter-action between individual S layers in the sample.27As a

re-sult, for certain values of superconducting layer thickness and of the thickness of interlayers separating them, the di-mensional crossover could be observed on the angular de-pendence Hc共⌰兲 when the 3D mode at ⌰=␲/ 2 changes smoothly into 2D dependence as the angle⌰ decreases to zero.12,28–30

Apart from this aspect other theoretical attempts have been done to describe the whole Hc共⌰兲 curve mainly for

infinite multilayers of different coupling nature between S layers.31–36On the other hand, the finiteness of the

multilay-ers samples共i.e., the precise position of the superconducting nucleus兲 has been found to be a very important parameter in determining, for example, the resistive transition to the su-perconducting state16 and the shape of the H-T phase

diagram.17 From this viewpoint the survey of the Hc共⌰兲

characteristics in multilayers with the possibility of fine-tuning the position of superconducting nucleus is of great significance at least for two reasons: first, because of the mutual relation between the Hc共⌰兲 shape and the nucleation position of the superconducting phase and, second, because in multilayers it is possible to precisely control, change, and know the position where the superconducting phase nucle-ates.

In this work we present the results of measurements of the Hc共⌰兲 dependences for Nb/Pd multilayers at different tem-peratures and we analyze the difference with the existing theoretical descriptions in order to deepen the understanding of the nature of the proximity coupling and of the wave function distribution in S/N multilayered nanostructures. The Nb/ Pd system was chosen because of its pronounced inter-face transparency.37We prepared Nb/ Pd samples with differ-ent geometrical symmetries in order to vary the position of the nucleation of the superconducting phase.17 The angular

measurements were performed on the same set of samples analyzed in our previous work in which we reported only the results of the Hc temperature dependence.17 We performed

angular measurements in the temperature range where, ac-cording to the Hc共T,⌰=0兲 result, samples are in 2D mode.

However, in agreement with previous experimental works,4,25,28the measured H

c共⌰兲 data were not described by

the 2D Tinkham expression关Eq. 共2兲兴. We show that, at fixed temperature, for the sample with the symmetry plane in the central S layer the 2D mode remains up to higher⌰ values with respect to the characteristic of the sample in which the symmetry plane falls in the central N layer. We relate this

fact to the more pronounced bidimensionality in the former case as was discussed in Refs. 16 and 17. We discuss the experimental method for the determination of the angle⌰*, the angle where the dimensional crossover occurs, and we estimate the Hc共⌰兲 dependence for finite S/N multilayers in

the simplest way within the GL approach.

EXPERIMENT

The Nb/ Pd samples were grown on Si共100兲 substrates at room temperature by using a dual-source magnetically en-hanced dc triode sputtering system with a movable substrate holder.11 The high-quality layered structure of the samples, with negligibly small interfacial roughness共order of 1 nm兲, was confirmed by high- and low-angle x-ray diffraction measurements.11 A specially designed movable shutter

al-lowed the simultaneous deposition of the two samples with a different number of bilayers, Nbil.17 We have fabricated

Nb/Pd samples in the same deposition run with Nbil= 9 and

10, respectively. The relation between the thickness of Pd 共dN兲 and Nb 共dS兲 layers was dN= dS/ 2 = 10 nm. For both

samples the top and bottom layers were made of the normal layer. This means that the sample with Nbil= 9共=10兲 consists

of 9 共10兲 Pd/Nb bilayers plus the capping Pd layer. As a result, the geometrical plane of the sample symmetry falls into the center of the central S layer for Nbil= 9共sample SP兲

and into the center of the central N layer for Nbil= 10共sample NP兲.17The fabrication of two samples in the same deposition

run allows us to assume the same Nb and Pd properties in both samples as well as the same interface transparency, giv-ing us the possibility to suppose geometrical symmetry as the only difference between the two systems.

We performed dc transport measurements with a standard four-probe technique with a variable orientation of the exter-nal magnetic field which was created by a superconducting solenoid. The accuracy of the rotation angle was ±0.1°. The Hcvalues were extracted from the R共H兲 curves measured at the onset of the superconducting transition. The temperature stabilization during the measurements was ±0.01 K. More details about the properties of these samples were reported elsewhere.11,17

RESULTS

In Ref. 17 we published the results of a detailed study of the H-T phase diagrams for these samples. We established that the sample NP reveals the classical Hc共T,⌰=0兲

depen-dence for S/N multilayers having dS⬃dN⬃␰—i.e., the

pres-ence of temperature dimensional crossover at temperature T*= 3.60 K 关Fig. 2共a兲 of Ref. 17兴. At T⬍T*, Hc共T,⌰=0兲

⬃共Tc2D− T兲1/2共Tc2Dbeing the critical temperature in the 2D

mode兲, which is a signature of bidimensionality, while at T ⬎T* multilayer behaves like a 3D system and H

c共T,⌰=0兲

⬃共Tc− T兲.1On the other hand, sample SP did not reveal the

pronounced linear part in the Hc共T,⌰=0兲 dependence 关Fig. 2共b兲 of Ref. 17兴 and the Hc共T,⌰=0兲 curve was square root like in the whole temperature range, even close to Tc. Such a difference is related to the different symmetries of the order parameter describing the two systems. At T close to Tcin the

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sample SP the superconducting nucleus appears in the central S layer 共the nucleation position is z0= 0兲 and this situation

remains unchanged in the entire temperature range 0⬍T ⬍Tc. For the sample NP for T*⬍T⬍Tcthe superconducting

nucleus is spread between two central S layers while for T ⬍T*it is shifted to one of the S layers adjacent to the central

N layer, changing the position of the nucleation from z0= 0 共at T*⬍T⬍T

c兲 to z0= ±⌳/2 共at T⬍T*兲. Here ⌳=dS+ dNis

the period of the layered structure. This change of symmetry of the order parameter in the sample NP causes a sharp 2D-3D crossover.17

In this paper we first show the results for Hc共⌰兲 for the sample NP. We performed angular measurements in the tem-perature range 1.8 K⬍T⬍T*. In Fig. 1 such characteristics

measured at three fixed temperatures T = 3.45 K, 2.91 K and 2.05 K are presented. The general features of all curves is the presence of a cusp at⌰=0. This behavior has been obtained at all the other measured temperatures. At first glance, this result is a consequence of the bidimensional behavior of Hc共T,⌰=0兲 at T⬍T*, but Tinkham’s expression共2兲 does not

fit the experimental data 共dashed curves in Fig. 1兲. As was shown for S/I superlattices,27,28 the reason is the incorrect

application of Eq.共2兲 for describing the whole Hc共⌰兲

depen-dence. In fact, the Hc共⌰兲 curve is divided into two parts

which are related to two different dimensionalities of the sample even at T⬍T*. In other words, an angle * exists

such that, when⌰⬍⌰*, the experimental dependence is well

described by Tinkham’s expression in which, however, the Hc共␲/ 2兲 value is a free parameter and it is smaller than the

measured value of the perpendicular critical magnetic field. In our case this fit is shown for each temperature as a solid line in Fig. 1. For⌰⬎⌰*, in the case of S/I superlattices, the

3D mode takes place and the angular dependence of Hc is described by Eq. 共1兲.28 The 3D-2D angular crossover

hap-pens in this case smoothly. In our sample, as is seen from Fig. 1, the situation is different. At T = 3.45 K a sharp transi-tion takes place at the angle⌰*. Actually the same is valid

also for T = 2.91 K and T = 2.05 K. Another peculiarity of this sample is related to the unusual 3D mode at⌰⬎⌰*. For the sake of convenience we show the Lawrence-Doniach fit关Eq. 共1兲兴 only for one temperature 关short-dotted line in Fig. 1共a兲兴. As can be seen this curve corresponds well to the experimen-tal data only in the tail of the curve when⌰ approaches␲/ 2. At the same time when using Eq.共2兲 with Hc共0兲 as a free

parameter we obtain very good correspondence to the experi-mental data关dash-dotted line in Fig. 1共a兲兴. The physical rea-son for this result will be discussed later.

In Fig. 2 we show the Hc共⌰兲 curves measured at three temperatures T = 3.50, 2.50, and 1.91 K for sample SP. By the dashed lines we show the result of Tinkham’s expression 共2兲, by the solid line the result of the Tinkham fit, but with Hc共␲/ 2兲 as a free parameter, and by the dash-dotted line the

result of the Tinkham fit with Hc共0兲 as a free parameter. As

follows from Fig. 2, the sample SP also presents the angular crossover, but the sharp change at⌰*does not arise. Another

peculiarity of this sample is the noticeably larger⌰* values and the smaller deviation of the experimental data from the Tinkham curve共2兲 obtained without fitting parameters. Both these facts indicate that sample SP, with the symmetry plane

in the center of the middle S layer, is “more” bidimensional than sample NP also from the viewpoint of the Hc共⌰兲 depen-dence, in agreement with the Hc共T,⌰=0兲 result.17

In Fig. 3 we plot the relative deviation of the experimental data from Eq.共2兲, ␦=关Hc共⌰兲/Hc2D共⌰兲兴2, as a function of ⌰

共Ref. 38兲 for sample NP. When the magnetic field is directed FIG. 1. Angular dependences of the upper critical magnetic field for sample NP. See text for the meaning of the different lines. Ar-rows indicate the value of⌰*for each temperature:共a兲 T=3.45 K, 共b兲 T=2.91 K, and 共c兲 T=2.05 K.

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along the basal plane of the substrate or along the perpen-dicular direction, ␦= 1, since the experimental values of Hc共0兲 and Hc共␲/ 2兲 were used as parameters for Eq. 共2兲. The shape of the deviation as a function of the angle is similar for all temperatures, but the angle⌰max, at which the deviation

⌬␦共⌰兲=1−␦共⌰兲 reaches its maximum value, is temperature dependent.

In Fig. 4 by open 共solid兲 symbols we show the ⌰* and ⌰maxtemperature dependences for the sample SP共NP兲. As is

seen, the values of⌰maxwell correspond to the values of⌰*.

This result indicates that the analysis of the␦ versus⌰ plot could be a simple method for finding out⌰*⬅⌰max.

DISCUSSION

The obtained 2D mode at⌰⬍⌰*for sample NP naturally fits the physical picture proposed in Refs. 27 and 28 and is related to the coherence length values in the temperature range 2.0 K , . . . , 3.6 K. Indeed, from the Hc共T,⌰=␲/ 2兲

de-pendence we may obtain the GL coherence length at zero temperature ␰共0兲,39 which in our case is equal to 12.6 nm.

From this we get that␰共T兲=共0兲共1−T/Tc兲−1/2changes from

⬇18 nm at T=2 K up to ⬇33 nm at T=3.60 K—i.e.,

⬃dSand␰艋⌳. This means that for T⬍T*the wave function

FIG. 2. Angular dependences of the upper critical magnetic field for sample SP. See text for the meaning of the different lines. Ar-rows indicate the value of⌰*for each temperature:共a兲 T=3.50 K, 共b兲 T=2.50 K, and 共c兲 T=1.91 K.

FIG. 3. Relative deviation␦=关Hc共⌰兲/Hc

2D共⌰兲兴2as a function of ⌰ for sample NP at three different temperatures. Arrows indicate the values of⌰maxfor each temperature.

FIG. 4. Temperature dependences of ⌰* and

max for sample NP共solid symbols兲 and for sample SP 共open symbols兲.

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of the superconducting condensate weakly changes with the thickness of the S layer, which is a signature of the 2D nature of superconductivity. When the sample is rotated from ⌰ = 0 towards⌰*the superconducting nucleus is still localized in one period of the S/N structure and the S/N interfaces favor the localization of the superconducting nucleus. For ⌰⬎⌰* two factors influence the H

c共⌰兲 curves. First, the

perpendicular component of the external magnetic field spreads the localization degree of the order parameter on more than one period. In this case the sample could not be concerned as 2D anymore. Second, the finite thickness of the sample determines the symmetry plane which falls in the middle of the central N layer. Consequently, in a perpendicu-lar field the order parameter is symmetric.17Due to this and

due to␰ values, the superconducting nucleus in the perpen-dicular field can be represented as dumbbell shaped, with two maxima in the adjacent central S layers.40 This is the

characteristic feature of 3D objects as also shown by linear Hc共0兲 versus T dependence17; but Eq.共1兲 is not valid in this

case because it was deduced in the approximation of the homogeneous infinite medium. For our multilayers the super-conducting order parameter, changing significantly in the whole sample, remains relatively homogeneous in one S layer. This probably explains the applicability of the Tinkham formula 共2兲 also at ⌰⬎⌰*. So we get, from one

side, at ⌰⬎⌰* an inhomogeneous structure with bound S

layers共i.e., 3D structure兲, and from the other, at ⌰⬍⌰*, we

get an analogous of the homogeneous thin film. ⌰* is the angle where this change of topology of the superconducting nucleus takes place.

Within the above-described qualitative picture one can es-timate the angular dependences of the upper critical field for finite S/N metal multilayers taking account of the nucleation position of the superconducting phase. Here we interpret the experimental results by estimating Hc共⌰兲 within the simplest

variation of the GL model,20–22,41 taking account of the real

symmetry of the multilayers.

We chose the axis coordinates in the following way: the axis OZ is directed perpendicular to the layer surface and the coordinate plane XOY is parallel to layers and coincides with the symmetry plane of the multilayer. The superconductor structure is supposed to be infinite along the OX and OY directions and has thickness L along the OZ direction. The vector potential A共r兲⬅共Hz cos ⌰−Hy sin ⌰,0,0兲 is related to the external magnetic field H⬅共0,H cos ⌰,H sin ⌰兲. Taking into account that in the critical region the GL wave function could be expressed as ⌿共r兲=eikx共y,z兲, we may write the GL functional in the following way:

F =

冕冕

dydz兵共⳵y␺兲2+共⳵z␺兲2− a共z,T兲␺2

+ H2共z cos ⌰ − z0− y sin⌰兲2␺2其, 共3兲

where z0⬅k/H0; a共z;T兲 is the step function.18We minimize

Eq.共3兲, by choosing the probe wave function with separated variables:39

共y,z兲 →共y,z兲 = f共y兲g共z兲, 共4兲 with the boundary conditions f共±⬁兲=0 and

g

共±L/2兲 = 0. 共5兲

Moreover, on S/N interfaces the join condition has to be fulfilled,1,42

g

g

zi+0 = pi

g

g

zi−0 , 共6兲

where the quantities pidetermine the jump of the logarithmic

derivative of the wave function. The expression for g共z兲 would be

g

共z兲 + 关a共z,T兲 − H2cos2⌰共z − z¯兲2− H兩sin ⌰兩兴g共z兲 = 0, 共7兲 where z¯ means the average over the distance. By solving Eqs.共5兲–共7兲 one can find the required Hc共⌰兲 dependence.

Moreover, from Eqs.共3兲 and 共4兲 it is possible to extract the expression for the important quantity ␣共⌰兲 ⬅Hc−1共⌰兲dHc共⌰兲/d⌰, whose analytical properties at ⌰=0

determine the effective dimension of the superconducting nucleus. In fact, for⌰=0 we get

共⌰ = ± 0兲 = − sgn共⌰兲/关Hc共0兲␴2兴, 共8兲

where␴2⬅兩共z−z¯兲2兩⌰=0.

From Eq.共8兲 it follows that within the approach used the relative drop of the Hc共⌰兲 characteristic at ⌰=0 is

deter-mined only by the parameter of localization of the wave function␴in the region of z¯.

In order to check qualitatively the physical meaning of the model we performed the calculation of Hc共⌰兲 for S/N

mul-tilayers with even Nbil. We chose Nbil= 4 共plus a capping N

layer兲 as a model object. We also performed the simulations for NSN trilayer共i.e., one S/N bilayer兲 which is considered analogous to thin films. We assume dN= 0.5dS=␰S共0兲

= 2␰N共Tc兲. Here␰S共N兲is the value of the coherence length in

the S共N兲 layer. The temperature dependences of the coher-ence lengths were assumed as ␰S共T兲=␰S共0兲共1−T/TS兲−1/2

共Ref. 39兲 and ␰N共T兲=␰N共Tc兲共T/Tc兲−1/2 共Ref. 43兲 共Ts is the

critical temperature of the bulk superconducting material兲. In Fig. 5 we show the calculated Hc共⌰兲 characteristics at T

close to T*. The result for the one bilayer sample exactly reproduces Tinkham’s curve for a 2D homogeneous super-conductor. In this case there is only one parameter z¯ = 0 that corresponds to the upper critical field in the whole tempera-ture range. The function g共z兲 is symmetrical. At ⌰=0 the superconducting nucleus is situated in the symmetry plane of the sample. For the four bilayer structure the 2D behavior with function g共z兲 localized in one S layer realizes for ⌰ ⬍⌰* 共open circles in Fig. 5兲; at ⌰⬎⌰* the function g共z兲

spreads over two S layers and the Hc共⌰兲 behavior changes to 3D one共solid circles in Fig. 5兲.

The obtained results of numerical simulations illustrates the angular dimensional crossover and the influence of the nucleation position共parameter z¯兲 and of the value of local-ization parameter␴on the Hc共⌰兲 shape. Indeed, the 2D be-havior corresponds to␴艋dSand z¯ falls into the center of one of S layers.16From the experimental data of Figs. 1 and 2 we

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to Eq.共8兲 and we have obtained 2␴⬇16 nm for sample SP 共at T=1.91 K兲 and 2⬇14 nm 共at T=2.05 K兲 for sample NP, close to the Nb thickness共20 nm兲. Note that the values of 2␴ are also in very reasonable agreement with the

共T兲 values extracted from the Hc共T,⌰=␲/ 2兲 dependences.

关For NP ␰共T=2.05 K兲⬇18 nm and for SP共T=1.91 K兲 ⬇19 nm兴. So the calculated results reproduce the main fea-tures of the experimental characteristics.

The positions of the nuclei of the superconducting phase are in close relation to the character of Hc共⌰兲 dependences, as follows from the analysis of the data for our even and odd Nb/ Pd multilayers. For finite S/N multilayers the position of the nucleus,具z典⌰=0= z0, accesses one of the discrete available

values corresponding to the upper critical magnetic field. At low temperatures 共TⰆTc兲 the values of the magnetic field

which correspond to different positions of nuclei are almost equal; i.e., nuclei of the superconducting phase are confined with almost the same probability in one of the superconduct-ing layers both for nine- and ten-bilayer structures.17And as

our estimations show, the wave function decays relatively quickly within the single S layer. So the multilayer structure is actually reduced to one slab. It is clear that in this tem-perature range for both odd and even S/N samples the same Hc共⌰兲 curves should be observed. In Fig. 6 we show the

Hc共⌰兲 dependences for both samples measured at T

= 1.91 K 共sample SP兲 and at T=2.05 K 共sample NP兲. It is seen that both curves are matched well.

Within the proposed physical picture it is also possible to explain the smaller Hc共␲/ 2兲 values obtained from the fitting

procedure using Eq. 共2兲 at ⌰⬍⌰*. Indeed, in this angular range, our sample is reduced to a simple one-bilayer S/N structure with smaller Tc value with respect to the 3D

multilayer structure 共which has larger total S layer thickness兲.42 The reduction of * with temperature共Fig. 4兲

can also be explained within the proposed description. In fact, while increasing the temperature␰共T兲 increases and this reduces the localization degree of the superconducting nucleus in a parallel field, causing its delocalization at lower ⌰*values. This delocalization at smaller*, finally, causes a

larger change of the derivative. Moreover, the change of the derivative dHc共⌰兲/d⌰ is sharper at ⌰* for sample NP and

this is explained by the sudden change of the nucleation position of the superconducting phase which in sample SP does not actually occur.

CONCLUSION

In conclusion, we have conducted a systematic study of the angular dependences of the upper critical magnetic field for Nb/ Pd multilayers with different geometrical symme-tries. Samples with an odd and even number of N/S bilayers 共plus capping N layer兲 showed the change of dimensionality with the angle of application of the external magnetic field. In the range 0⬍⌰⬍⌰* samples are reduced to thin-film slabs 共i.e., they are two dimensional兲 while for angles ⌰*

⬍⌰⬍␲/ 2 samples must be considered as actual multilayers 共three dimensional兲. The crossover angle ⌰* is temperature

dependent due to the temperature dependence of the super-conducting coherence length.

The position of the superconducting nucleus influences the Hc共⌰兲 shape, revealing a stronger bend for a multilayer

with an odd number of bilayers. This reflects the change of the nucleus position between⌰=0 and ⌰=␲/ 2. At the same time for a multilayer with an even number of bilayers the nucleus position is localized at the geometrical center of the sample at both⌰=0 and ⌰=␲/ 2, and this is reflected in a larger bidimensionality of the sample. Based on such a physical picture and using the GL approach, an expression for Hc共⌰兲 was derived which seems to reasonably explain

the presence of angle dimensional crossover, the measured characteristics at small⌰ for different temperatures, the po-sitions of the superconducting nucleus, and the symmetry of the S/N multilayers.

FIG. 5. Calculated Hc共⌰兲 dependence for one bilayer 共open squares兲 and four bilayer 共circles兲 structures at T艋T* 共see text兲. Open共solid兲 circles correspond to 2D 共3D兲 behavior. The solid line is built according to Tinkham’s expression for a thin film. Hcsis the critical magnetic field of the bulk material.

FIG. 6. Hc共⌰兲 for sample SP measured at T=1.91 K and for sample NP measured at T = 2.05 K.

(7)

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Figura

FIG. 2. Angular dependences of the upper critical magnetic field for sample SP. See text for the meaning of the different lines
FIG. 6. H c 共⌰兲 for sample SP measured at T=1.91 K and for sample NP measured at T = 2.05 K.

Riferimenti

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